Barycentric Hermite Interpolation

TL;DR

This paper presents an efficient, stable barycentric Hermite interpolation method that allows quick updates of weights when additional derivatives are prescribed, improving computational efficiency and stability over previous approaches.

Contribution

The authors derive a new, efficient method for updating barycentric weights in Hermite interpolation, enabling quick adjustments with minimal computational effort.

Findings

Allows updating weights with O(N) operations when adding derivatives.

Provides a numerically stable method suitable for high-order derivatives.

Achieves fewer operations than existing methods for barycentric weight computation.

Abstract

Let $z_{1},\ldots,z_{K}$ be distinct grid points. If $f_{k,0}$ is the prescribed value of a function at the grid point $z_{k}$, and $f_{k,r}$ the prescribed value of the $r$\foreignlanguage{american}{-th} derivative, for $1\leq r\leq n_{k}-1$, the Hermite interpolant is the unique polynomial of degree $N-1$ ($N=n_{1}+\cdots+n_{K}$) which interpolates the prescribed function values and function derivatives. We obtain another derivation of a method for Hermite interpolation recently proposed by Butcher et al. {[}\emph{Numerical Algorithms, vol. 56 (2011), p. 319-347}{]}. One advantage of our derivation is that it leads to an efficient method for updating the barycentric weights. If an additional derivative is prescribed at one of the interpolation points, we show how to update the barycentric coefficients using only $\mathcal{O}\left(N\right)$ operations. Even in the context of confluent Newton series, a comparably efficient and general method to update the coefficients appears not to be known. If the method is properly implemented, it computes the barycentric weights with fewer operations than other methods and has very good numerical stability even when derivatives of high order are involved. We give a partial explanation of its numerical stability.